A well-known formula of Tutte and Berge expresses the size of a maximum matching in a graph in terms of what is usually called the deficiency of . A subset of for which this deficiency is attained is called a Tutte set of . While much is known about maximum matchings, less is known about the structure of Tutte sets. In this article, we study the structural aspects of maximal Tutte sets in a graph . Towards this end, we introduce a related graph . We first show that the maximal Tutte sets in are precisely the maximal independent sets in its -graph , and then continue with the study of -graphs in their own right, and of iterated -graphs. We show that is isomorphic to a spanning subgraph of , and characterize the graphs for which and for which . Surprisingly, it turns out that for every graph with a perfect matching, . Finally, we characterize bipartite -graphs and comment on the problem of characterizing -graphs in general.